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Test driven programming
What is it?
Why does it matter in P3?
Additional help desk hours
4-8 Sundays
4-6 Mondays
12-6 Thursdays
10-3 Fridays
Iclicker Question #1
What is the definition of recursion?
A. See recursion
Recursion
Chapter 5.4 in Rosen
Chapter 11 in Savitch
Preliminaries: programming as a contract
Specifying what each method does
Specify it in a comment before method's header
Precondition
What is assumed to be true before the method is
executed
Caller obligation
Postcondition
Specifies what will happen if the preconditions are
met – what the method guarantees to the caller
Method obligation
Example
/*
** precondition: n >= 0
** postcondition: return value satisfies:
** result == n!
*/
int factorial(int n) {
}
Enforcing preconditions
/*
** precondition: n >= 0
** postcondition: return value satisfies:
** result == n!
*/
int factorial(int n) {
if (n < 0)
throw new ArithmeticException(“cannot
compute factorial of a neg number!”);
}
What about postconditions?
/*
** precondition: n >= 0
** postcondition: return value satisfies:
** result == n!
*/
int factorial(int n) {
assert result == n!;
}
Java provides a mechanism called assertions to verify that
postconditions hold
Enforcing preconditions
/*
** precondition: x >= 0
** postcondition: return value satisfies:
** result * result == x
*/
double sqrt(double x) {
if (x < 0)
throw new ArithmeticException(“you
tried to take sqrt of a neg number!”);
}
What does this method do?
/**
* precondition n>0
* postcondition ??
*/
private void printStars(int n) {
if (n == 1) {
System.out.println("*");
} else {
System.out.print("*");
printStars(n - 1);
}
}
Recursion
recursion: The definition of an operation in terms
of itself.
Solving a problem using recursion depends on solving
smaller occurrences of the same problem.
recursive programming: Writing methods that
call themselves
directly or indirectly
An equally powerful substitute for iteration (loops)
But sometimes much more suitable for the problem
Recursive Acronyms
http://search.dilbert.com/comic/Ttp
Dilbert: Wally, would you like to be on my TTP project?
Wally: What does "TTP" stand for?
Dilbert: It's short for The TTP Project. I named it myself.
— Dilbert, May 18, 1994
GNU — GNU's Not Unix
KDE — KDE Desktop Environment
PHP - PHP: Hypertext Preprocessor
PNG — PNG's Not GIF (officially "Portable Network Graphics")
RPM — RPM Package Manager (originally "Red Hat Package
Manager")
More Recursion!
http://xkcd.com/688/
http://xkcd.com/981/
Why learn recursion?
A different way of thinking about problems
Can solve some problems better than
iteration
Leads to elegant, simple, concise code (when
used well)
Some programming languages ("functional"
languages such as Scheme, ML, and
Haskell) use recursion exclusively (no loops)
Exercise
(To a student in the front row)
How many students are directly behind you?
We all have poor vision, and can
only see the people right next to us.
So you can't just look back and count.
But you are allowed to ask
questions of the person behind you.
How can we solve this problem?
(recursively )
The idea
Recursion is all about breaking a big problem
into smaller occurrences of that same problem.
Each person can solve a small part of the problem.
What is a small version of the problem that would be easy to
answer?
What information from
a neighbor might help
you?
Recursive algorithm
Number of people behind me:
If there is someone behind me,
ask him/her how many people are behind him/her.
When they respond with a value N, then I will answer N + 1.
If there is nobody behind me, I will answer 0.
Recursive structures
A directory has
files
and
(sub) directories
An expression has a*b + c*d
operators
operands, which are
variables
constants
(sub) expressions
Expressions represented by trees
A tree is
a node
with
zero or more sub
trees
examples:
a*b + c*d
(a+b)*(c+d)
+
* *a b c d
*+ +
a b c d
Structure of recursion
Each of these examples has
recursive parts (directory, expression, tree)
non recursive parts (file, variables, nodes)
Always need non recursive parts. Why?
Same goes for recursive algorithms.
Cases
Every recursive algorithm has at least 2 cases:
base case: A simple instance that can be answered
directly.
recursive case: A more complex instance of the
problem that cannot be directly answered, but can
instead be described in terms of smaller instances.
Can have more than one base or recursive case, but all
have at least one of each.
A crucial part of recursive programming is identifying
these cases.
Base and Recursive Cases: Example
public void printStars(int n) {
if (n == 1) {
// base case; print one star
System.out.println("*");
} else {
// recursive case; print one more star
System.out.print("*");
printStars(n - 1);
}
}
Recursion Zen
An even simpler, base case is n=0:
public void printStars(int n) {
if (n == 0) {
// base case; end the line of output
System.out.println();
} else {
// recursive case; print one more star
System.out.print("*");
printStars(n - 1);
}
}
Recursion Zen: The art of identifying the best set
of cases for a recursive algorithm and expressing
them elegantly.
Everything recursive can be done non-
recursively// Prints a line containing a given number of stars.
// Precondition: n >= 0
public void printStars(int n) {
for (int i = 0; i < n; i++) {
System.out.print("*");
}
System.out.println();
}
IC Question #1
What is a precondition for a method?
A. Method obligation
B. What is assumed to be true after the method is
executed
C. Caller obligation
D. The digital signature
25
IC Question #1 Answer
What is a precondition for a method?
A. Method obligation
B. What is assumed to be true after the method is
executed
C. Caller obligation
D. The digital signature
26
IC Question #2
What is the base case of a recursive
method?
A. A simple instance that can be answered
directly.
B. A more complex instance of the problem that
cannot be directly answered, but can instead
be described in terms of smaller instances.
C. The optional part of the recursive method
27
IC Question #2 Answer
What is the base case of a recursive
method?
A. A simple instance that can be answered
directly.
B. A more complex instance of the problem that
cannot be directly answered, but can instead
be described in terms of smaller instances.
C. The optional part of the recursive method
28
IC Question #3
What is the recursive case of a recursive
method?
A. A simple instance that can be answered
directly.
B. A more complex instance of the problem that
cannot be directly answered, but can instead
be described in terms of smaller instances.
C. The optional part of the recursive method
29
IC Question #3 Answer
What is the recursive case of a recursive
method?
A. A simple instance that can be answered
directly.
B. A more complex instance of the problem that
cannot be directly answered, but can instead
be described in terms of smaller instances.
C. The optional part of the recursive method
30
Let’s look at some code
SimpleRecursion
IC Question #1
What is a postcondition for a method?
A. Method obligation
B. What is assumed to be true before the method is
executed
C. Caller obligation
D. The digital signature
32
IC Question #1 Answer
What is a postcondition for a method?
A. Method obligation
B. What is assumed to be true before the method is
executed
C. Caller obligation
D. The digital signature
33
Computing the maximum of an array
We would like to compute the maximum of the
elements in an array.
Need to identify base case and recursive case,
and what to do in each.
Computing the maximum of an array
The idea:
Base case: array of size 1: return the element
Recursive case: compute maximum of two
numbers – the first element, and the maximum
of the rest of the array
Exercise
Let’s write a method reverseLines(Scanner scan)
that reads lines using the scanner and prints them in
reverse order.
Use recursion without using loops.
Example input: Expected output:
this no?
is fun
fun is
no? this
What are the cases to consider?
How can we solve a small part of the problem at a time?
What is a file that is very easy to reverse?
Reversal pseudocode
Reversing the lines of a file:
Read a line L from the file.
Print the rest of the lines in reverse order.
Print the line L.
If only we had a way to reverse the rest of the lines of the
file....
Reversal solution
public void reverseLines(Scanner input) {
if (input.hasNextLine()) {
// recursive case
String line = input.nextLine();
reverseLines(input);
System.out.println(line);
}
}
Where is the base case?
Call stack: The method invocations active at
any given time.
reverseLines(scanner);
input file:output:
thisisfunno?
no?funisthis
Tracing our algorithm
public void reverseLines(Scanner input) {if (input.hasNextLine()) {
String line = input.nextLine(); // ”this"reverseLines(input);System.out.println(line);
}}
public void reverseLines(Scanner input) {if (input.hasNextLine()) {
String line = input.nextLine(); // ”is"reverseLines(input);System.out.println(line);
}}
public void reverseLines(Scanner input) {if (input.hasNextLine()) {
String line = input.nextLine(); // ”fun"reverseLines(input);System.out.println(line);
}}
public void reverseLines(Scanner input) {if (input.hasNextLine()) {
String line = input.nextLine(); // ”no?"reverseLines(input);System.out.println(line);
}}
public void reverseLines(Scanner input) {if (input.hasNextLine()) { // false
...}
}
40
Recursive power example
Write a method that computes xn.
xn = x * x * x * ... * x (n times)
An iterative solution:
public int pow(int x, int n) {
int product = 1;
for (int i = 0; i < n; i++) {
product = product * x;
}
return product;
}
Exercise solution
// Returns base ^ exponent.
// Precondition: exponent >= 0
public int pow(int x, int n) {
if (n == 0) {
// base case; any number to 0th power is 1
return 1;
} else {
// recursive case: x^n = x * x^(n-1)
return x * pow(x, n-1);
}
}
42
How recursion works
Each call sets up a new instance of all the
parameters and the local variables
When the method completes, control returns to
the method that invoked it (which might be
another invocation of the same method)
pow(4, 3) = 4 * pow(4, 2)
= 4 * 4 * pow(4, 1)
= 4 * 4 * 4 * pow(4, 0)
= 4 * 4 * 4 * 1
= 64
43
Activation records Activation record: memory that Java allocates to store
information about each running method
return point ("RP"), argument values, local variables
Java stacks up the records as methods are called (call stack); a
method's activation record exists until it returns
Eclipse debug draws the act. records and helps us trace the behavior of a recursive method
_
| x = [ 4 ] n = [ 0 ] | pow(4, 0)
| RP = [pow(4,1)] |
| x = [ 4 ] n = [ 1 ] | pow(4, 1)
| RP = [pow(4,2)] |
| x = [ 4 ] n = [ 2 ] | pow(4, 2)
| RP = [pow(4,3)] |
| x = [ 4 ] n = [ 3 ] | pow(4, 3)
| RP = [main] |
| | main
44
Infinite recursion
A method with a missing or badly written base
case can causes infinite recursion
public int pow(int x, int y) {
return x * pow(x, y - 1); // Oops! No base case
}
pow(4, 3) = 4 * pow(4, 2)
= 4 * 4 * pow(4, 1)
= 4 * 4 * 4 * pow(4, 0)
= 4 * 4 * 4 * 4 * pow(4, -1)
= 4 * 4 * 4 * 4 * 4 * pow(4, -2)
= ... crashes: Stack Overflow Error!
An optimization
Notice the following mathematical property:
We can use it to improve our pow method!
What is the “trick”?
How can we incorporate this optimization into our pow method?
What is the benefit?
312 = (32 )6 = (9)6 = (81)3 =81*(81)2
Exercise solution 2
// Returns base ^ exponent.
// Precondition: exponent >= 0
public int pow(int base, int exponent) {
if (exponent == 0) {
// base case; any number to 0th power is 1
return 1;
} else if (exponent % 2 == 0) {
// recursive case 1: x^y = (x^2)^(y/2)
return pow(base * base, exponent / 2);
} else {
// recursive case 2: x^y = x * x^(y-1)
return base * pow(base, exponent - 1);
}
}
IC Question #2
In a recursive method, when the method
completes, control returns to where?
A. The calling method
B. The next iteration of the recursive method
C. The parent process
D. The child process
47
IC Question #2 Answer
In a recursive method, when the method
completes, control returns to where?
A. The calling method
B. The next iteration of the recursive method
C. The parent process
D. The child process
48
IC Question #3
What is an activation record?
A. Global memory
B. Memory specific to setting up and initializing a
program
C. Memory allocated for information about each
running method
D. The call stack
49
IC Question #3 Answer
What is an activation record?
A. Global memory
B. Memory specific to setting up and initializing a
program
C. Memory allocated for information about each
running method
D. The call stack
50
IC Question #4
When is an activation record released?
A. When the process terminates
B. When the method is invoked
C. When the process is created
D. When the method returns
51
IC Question #4
When is an activation record released?
A. When the process terminates
B. When the method is invoked
C. When the process is created
D. When the method returns
52
